Regularity and radial symmetry of positive solutions for a higher order elliptic system

We discuss the properties of solutions for the following elliptic partial differential equations system in R-n , {(-Delta)(alpha/2) u = u(p1) v(p2), (-Delta)(alpha/2) v = u(q1) v(q2), where 0 < alpha < n, p (i) and q (i) (i = 1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p (1) + p (2) = q (1) + q (2) = n+alpha/n-alpha, we classify the solutions of the PDEs system.